Long-distance order parameter

We have perfectly defined the two states of order and disorder. However, we can imagine intermediary states - e.g. a state where a certain number of atoms of A are on sites that are normally attributed to A in the perfectly ordered solution. This number would obviously be between the average and the total number of atoms of A. In order to characterize such an intermediary state, Bragg and Williams defined a degree of order or long-distance order parameter s, such that this degree is equal to 1 if the solution is perfectly ordered, and 0 in the case of a completely random distribution solution. 

However it should be pointed out that parameter ae, determined by long-range order r, concentrations a, b, and energetic parameters a, P, will be dissimilar for octa- and tetrahedral interstitial sites, because energies a and P of interatomic interaction depend on the distance between atoms and... 

Condensed matter phases and structures are commonly reached via symmetry breaking transitions. In such systems, when the continuous symmetry is broken, temporary domain-t5q)e patterns are formed. The domain structures eventually coarsen, and disappear in the long-time limit, leaving a uniform broken-symmetry state. This state possesses so-called long-range order (LRO), in which the spatially dependent order parameter correlation function does not decay to zero in the limit of large distances. 

An alternative picture was first introduced by Aharony and Pytte in the context of random magnets. In this picture the order parameter correlation function exhibits algebraic decay with distance instead. This situation, intermediate between SRO and LRO, has come to be known as quasi-long-range order (QLRO). The most well-known example of QLRO, due to Berezinsky and to Kosterlitz and Thouless occurs in the low temperature phase of the two-dimensional XY model. A number of recent theoretical and computational studies have supported this point of view in random spin systems in a higher dimensionality . 

Fig. 50. Magnetization profiles across thin Ising films [eq. (1) with H — H — 0], upper part, and near the surface of semi-infinite Heisenberg ferromagnels, lower part (where bulk behavior in the Monte Carlo simulation is enforced by an effective field boundary condition at z — 16). Note that in the Ising case (where three film thicknesses L = 5, 10, and 20 are shown) the surface layer magnetization m- — m(z — 0) is independent of L, and for L > 10 already the bulk value of the order parameter is reached in the center of the film. For the Heisenberg model, on the other hand, at a comparable temperature distance from % the free surface produces a Long-range perturbation of the local magnetization m(z). From Binder and Hohenbcrg (1974).

The initial modulation and the first peak in g(r) are due to the formation of local stmcture in the liquid. For completely uncorrelated systems, g(r) = 1, and thus the order parameter is zero. For a system with long-range order, the modulation in g(r) persists over large distances, causing the translational order to grow. 

Completely statistical (random) arrangements of the macromolecules without a regular order or orientation, i.e., without constant distances, are known as amorphous states. There is no long-range order whatsoever. The valid model for such states is the statistical coil. This is the dominating secondary structure in synthetic polymers and polymeric solutions. Its determinant parameter is coil density. 

For nematic liquid crystals, the synunetry is reduced and we need additional variables. The nematic is degenerate in the sense that all equilibrium orientations of the director are equivalent. According to the Goldstone theorem the parameter of degeneracy is also a hydrodynamic variable for a long distance process 0 and the relaxation time should diverge, x—>oo. In nematics, this parameter is the director n(r), the orientational part of the order parameter tensor. For a finite distortion of the director over a large distance (L—>oo), the distortion wavevector 0 and the... 

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